Selection Accounting: The Price Equation Runs the Economy

Two accountants, a century apart, wrote down the same identity

In 1970 the population geneticist George Price, working alone and largely outside any university appointment, published an exact identity for how the average value of any trait in any population changes from one generation to the next: partly because some types leave more descendants than others, and partly because each lineage’s own value shifts along the way ^[1]. A different literature had been assembling the same algebra without knowing it. Plant- and firm-level productivity economists, reading Census microdata rather than pedigrees, kept splitting aggregate productivity growth into a term for market-share reallocation and a term for improvement inside surviving firms — the tool now standard in every study of manufacturing dynamism, from Olley and Pakes’s telecommunications-equipment plants ^[6] to Foster, Haltiwanger, and Krizan’s synthesis of the whole approach ^[7]. Neither literature cited the other. Price’s own paper sat for years inside a biology-only citation network before later work recovered its full generality, while the productivity-decomposition literature grew up entirely inside labor and industrial-organization economics, with no particular occasion for anyone in it to open a 1970 issue of Nature. Disciplinary boundaries, not conceptual difficulty, kept the two apart. A new 49-page paper, “Selection Accounting: The Price Equation as the Fundamental Theorem of Evolutionary Economics,” does the term-by-term reconciliation and asks what follows once it is done.

This article is that paper’s public landing page, not the paper itself: the claim, the three equations that carry it, the headline numbers, and the honest limits, with a link to the full proofs. It is also the first release in a new series, The Evolutionary Economics Papers, ten installments planned in total.

Read the full paper (PDF, 49 pages)

The claim, in the paper’s own words

Here is the paper’s abstract, lightly trimmed for length but not for content:

Biology’s Price equation and empirical economics’ plant- and firm-level productivity decompositions are the same accounting identity, discovered twice and never reconciled. The covariance term that splits aggregate productivity growth into reallocation and within-unit improvement is, term for term, Price’s split between selection and transmission, and the standard continuer-entrant-exiter decomposition is that identity applied to one partition of the economy. Under replicator share dynamics with a linear selection gradient, the selection component of productivity growth equals selection intensity times the share-weighted variance of log productivity — an economic Fisher theorem. Under a neutral null in which share changes carry no information about productivity, the variance of the measured selection term is governed by market concentration: the Herfindahl index is the economy’s inverse effective population size, so measured selection in concentrated industries is cheap to manufacture from noise alone. Iterating the identity over nested levels turns cross-country misallocation measurement into a statement about which level of the economy is selecting and which has its selection suppressed. The paper closes with an evolvability index and five falsifiable predictions.

Four numbered propositions carry that claim, each committing to progressively less than a universal law of economic behavior: an identity, a conditional theorem, a multi-level extension, and a statistical null. A reader can accept the first and reject the last without contradiction.

Reallocation is Price’s covariance term wearing an economist’s name

Price’s identity, applied to firms with market shares $s_i$ and log-productivity values $z_i$, reads:

The change in mean log productivity splits into a covariance between a firm’s relative fitness — its share-growth factor $w_i$ — and its productivity, plus a share-weighted average of each firm’s own productivity change, weighted by how much it grew. The first term is selection: value reallocated toward whoever is already ahead. The second is transmission: value created inside units that already exist. This holds for any partition of any economy in any period. It is exact bookkeeping, not a behavioral model, and it carries no error term to hide behind.

Proposition 1 is that this identity, applied to the continuer/entrant/exiter partition of an economy, does not resemble the productivity-decomposition literature — it is that literature. Melitz and Polanec’s four-term dynamic extension ^[8], the Olley-Pakes cross-sectional covariance ^[6], and the Foster-Haltiwanger-Krizan within/between/entry/exit accounting ^[7] are the same algebra with firms relabeled as lineages and market share relabeled as reproductive share. The distinction is not academic: Melitz and Polanec’s own reworking of Slovenian manufacturing data from 1995 to 2000 shows that the older two-term treatment of entry and exit biases measured aggregate productivity growth by up to ten percentage points relative to their four-term correction ^[8], precisely because a two-term method has nowhere to put the weighting choice a dynamic economy forces on it. The paper also settles a further, decades-old ambiguity: the “cross term” that Foster-Haltiwanger-Krizan’s method cannot fully sign turns out to be exactly the gap between initial-share-weighted and realized-share-weighted transmission, a distinction the Price framework surfaces immediately because biology had already had to make the same weighting choice explicit. And it separates two objects the literature had been treating as one: the Olley-Pakes covariance prevailing in any single cross-section is a stock, the running balance of every past selection episode, while Price’s covariance in one period is the flow that period contributes to that balance.

Selection intensity times variance is a theorem, not a slogan

Add one falsifiable assumption — that market share evolves under replicator dynamics with a linear selection gradient linking a firm’s productivity advantage to its realized share growth — and Proposition 2 states an economic reading of Fisher’s fundamental theorem of natural selection ^[2]:

The selection component of the growth rate of mean log productivity equals a selection-intensity coefficient $\beta$ times the share-weighted variance of log productivity across firms. The qualitative move is not new: Metcalfe applied Fisher’s principle to competing firms under replicator dynamics in the 1990s ^[4], and Andersen used Price’s equation itself, by name, to split economic change into selection and innovation effects in 2004 ^[5]. What the paper adds is a discipline on $\beta$: because realized market selection acts on profitability rather than on physical output alone, per Foster, Haltiwanger, and Syverson’s finding that price and physical productivity pull in opposite directions across surviving plants ^[14], $\beta$ is never a constant of nature. It must be estimated market by market, and any estimate built on revenue-based productivity will confound genuine efficiency with market power. A third result, Proposition 3, iterates the same identity over nested levels of the economy and reads Hsieh and Klenow’s cross-country misallocation measurements ^[10] as a statement about which level is doing the selecting and which level has its selection suppressed — misallocation, on this reading, is not a residual but selection a higher level of aggregation is failing to let through. The persistence that scales $\beta$ into a real forecast is itself already measured: regressing a firm’s productivity on its own lagged value yields autoregressive coefficients on the order of 0.6 to 0.8 in the plant- and firm-level literature ^[9], the direct economic analogue of the narrow-sense heritability that scales Fisher’s original biological theorem.

The dispersion the paper reads is already on the books

None of this needed new data collection; the empirical decomposition literature had already measured the quantities the identity organizes, just without the organizing frame. Within four-digit U.S. manufacturing industries, the average gap in log total factor productivity between a plant at the 90th percentile and one at the 10th is 0.651, a ratio of 1.92 — the plant at the top makes almost twice the output of the plant at the bottom with the same measured inputs — and the same measure runs above 5:1 in China and India ^[9]. Hsieh and Klenow’s own reallocation counterfactual, equalizing marginal products to U.S.-observed dispersion, would raise manufacturing TFP by 30 to 50 percent in China and 40 to 60 percent in India ^[10]. The Olley-Pakes covariance term for labor productivity averages about 50 log points within U.S. manufacturing industries, versus 20 to 30 log points in Western Europe, versus close to zero at the start of Central and Eastern Europe’s transition to a market economy ^[11] — three economies, three institutional environments, three sizes of measured selection. Over a ten-year window, Foster, Haltiwanger, and Krizan’s canonical decomposition attributes roughly half to two-thirds of U.S. manufacturing productivity growth to within-plant improvement and 25 to 26 percent to net entry, with the remainder in reallocation among survivors ^[7]; Baqaee and Farhi’s general-equilibrium accounting finds reallocation toward high-markup firms responsible for about half of aggregate U.S. TFP growth from 1997 to 2015 ^[13]. And the population feeding all of this has been shrinking: the U.S. business startup rate fell from an average of 12.0 percent in the late 1980s to 10.6 percent just before the 2008 recession, then below 8 percent ^[12]. None of these figures are competing estimates of one underlying number; they are the same accounting identity applied to different countries, different partitions of the economy, and different decades, which is exactly why setting their magnitudes side by side is informative rather than merely convenient. Every one of these numbers already existed. What the paper supplies is the single equation that makes them comparable.

Concentration manufactures “selection” out of nothing

Every one of those covariance terms is a realized statistic computed on one historical sample path, and molecular evolution learned the hard way not to trust such statistics without a baseline: Kimura’s neutral theory showed that most measured molecular substitution is drift, not selection, and the field has required a null model ever since before crediting any pattern to adaptation. Proposition 4 supplies the economic equivalent. Assume market shares move by pure noise, uncorrelated with productivity — no firm is being selected for or against — and the expected value of the measured selection term is exactly zero. Its variance is not zero, and it has a closed form:

Under the empirically common case where productivity dispersion does not systematically favor large or small firms, that variance collapses to a clean statement: the noise in the selection term scales with the Herfindahl-Hirschman concentration index, an antitrust yardstick every industrial economist already computes as the sum of squared market shares, and the index’s reciprocal is the economy’s effective population size in exactly Wright’s population-genetic sense — the size of an idealized equal-share population that would generate the same drift. A hundred equal firms and one firm with half the market plus two hundred sharing the rest can have the same headcount and wildly different $N_e$.

The paper checks this with a seeded simulation rather than leaving it asymptotic. A competitive industry of 400 comparably sized firms (HHI = 0.0031, $N_e \approx 321$) and a concentrated industry of 40 Zipf-distributed firms (HHI = 0.1006, $N_e \approx 10$) were each run through 20,000 replicate neutral periods with no selection whatsoever operating. A reading of half a log point of apparent “selection” — the kind of magnitude an Olley-Pakes-style decomposition would report without hesitation as meaningful reallocation — occurred in 4.0 percent of neutral periods in the competitive industry and in 49.7 percent of neutral periods in the concentrated one. In an industry with an effective population of ten, roughly a coin flip’s worth of periods will show economically meaningful “selection” with none present. The paper adds a further caution for anyone tempted to reach for a textbook significance test on a covariance term like this: real firm growth rates are tent-shaped and fat-tailed rather than Gaussian, so a nominal five-percent test can carry a true false-positive rate several times that whenever the effective population is small, and bootstrap or simulated critical values are recommended over normal-theory ones in exactly those settings.

This bears directly on the declining-dynamism debate. U.S. business concentration has risen over the same decades that startup rates and job reallocation have fallen ^[12], and Proposition 4 says that falling $N_e$ mechanically widens the drift band that any measured selection term must clear before it means anything. An economy moving toward the concentrated end of the range above is not only changing its market structure; it is degrading the statistical power of every reallocation decomposition run on the resulting data, exactly when reallocation is the thing everyone most wants to measure precisely.

What is new here, and what is not

The paper is explicit that the qualitative bridge is prior art, not its own discovery. Nelson and Winter founded the modern research program treating firm routines as heritable units and market competition as selection on profitability ^[3]; Metcalfe applied Fisher’s principle to competing firms directly ^[4]; Andersen used the Price equation itself, by name, to partition economic change ^[5]. “Economics is a selection process” has been argued for decades. What the paper adds, and claims no more than, is the term-by-term algebraic bridge to the modern empirical decomposition literature’s own measured magnitudes; the drift null and its identification of the Herfindahl index with inverse effective population size; a multi-level reading of misallocation as suppressed selection; and the evolvability index proposed below.

Three limits deserve equal billing with the results. Firms are not literal replicators — they merge, split, and change strategy in ways no organism does, so the analogy is a formal one, not a claim about biological mechanism. Productivity measurement itself conflates technical efficiency with market power, since selection observed through revenue-based data acts on profitability, not on physical output alone ^[14]. And an identity, however exact, is bookkeeping: it can tell you that reallocation happened and how large it was, but it cannot by itself tell you that anything caused anything. The paper is explicit on this last point rather than letting a reader assume otherwise, engaging directly with the standard biological caution that an exact accounting identity is not, on its own, a causal model of anything — a discipline population genetics learned to enforce on itself decades ago and that evolutionary economics is only now inheriting.

Five ways this program can be proven wrong

The paper closes by proposing an evolvability index — selection intensity times productivity variance times persistence, plus a term for variance injected by entry — built entirely from quantities the decomposition literature already estimates, and states five predictions specific enough to fail:

1. Within narrowly defined industries, the variance of the measured selection term should rise with market concentration once baseline noise is controlled for; a null or negative relationship would falsify the drift mechanism outright.
2. The Olley-Pakes covariance level should track the cumulative history of the period-by-period selection term, so a deregulation or trade shock that moves one without eventually moving the other would sever the claimed stock-flow link.
3. Diffusion of better management practice should raise the within-firm transmission share of productivity growth rather than the reallocation share, since better management changes firms rather than reallocating between them.
4. Higher-entry economies or periods should show wider cross-sectional productivity dispersion and, holding selection intensity fixed, larger measured selection components; compressed dispersion under high entry would refute the channel.
5. If the multi-decade decline in U.S. startup and reallocation rates reflects genuinely reduced selection rather than merely reduced turnover, decompositions run on recent data should show a smaller reallocation share and a larger within-firm share than the 1977–87 benchmark; a stable or rising reallocation share despite the documented decline in dynamism would refute the link.

The paper closes, the series is one paper old

None of the four propositions above requires believing that markets are kind, that concentration is always bad, or that competition should be read as combat. The claim is narrower and, for being narrower, more checkable: an accounting identity that biology and economics each wrote down independently, a conditional theorem for when it behaves like Fisher’s, and a null model for when it does not behave like anything at all. Selection Accounting is paper one of ten planned in The Evolutionary Economics Papers series; the next nine will each take one further piece of population thinking and force it through the same discipline — an identity first, a theorem only where the assumptions earn it, and a null before any claim of selection is allowed to stand.

Read the full paper (PDF, 49 pages)